Counting distinct diacritics on the Wikipedia page Diacritic , and ignoring the distinction between diacritics that generate new letters and diacritics that don’t:
- Vietnamese has nine: horn, circumflex, breve, bar (đ), acute, grave, tilde, underdot, and hoi (mini-question mark)
- Livonian has six (macron, umlaut, ogonek, superdot, tilde, hacek), but wins points for multiply stacked diacritics, like Vietnamese: ā, ä, ǟ, ḑ, ē, ī, ļ, ņ, ō, ȯ, ȱ, õ, ȭ, ŗ, š, ț, ū, ž. Livonian however is either moribund, extinct, or under revival.
- Lithuanian has four basic diacritics (caron, ogonek, macron, superdot); dictionaries also use acute, grave, tilde for pitch accent. So seven, though in practice only four.
- No others in the list have more than six.
So for commonplace Roman alphabets, Vietnamese still wins. Other scripts do better: Hebrew have 13 Niqqud, though of course vowel pointing is not a regular part of Hebrew orthography.
Minority languages with orthographies devised by modern linguists may have more diacritics. Though I suspect they don’t.
If phonetic alphabets count, then the IPA has at least 43 diacritics (depending on how you count them), and other phonetic alphabets are probably even more profligate.